In addition to Shannon diversity, reported in the manuscript, we conducted the analysis using richness as our focal diversity metric. During the with seed inflow stage, external seed addition is added to the simulated plant communities. This external seed addition is calculated to be 100% of the average monoculture seed production across all species. Therefore, even rare species are able to maintain low population sizes. For this reason, there is often no slope within each planted species richness treatment. In other words, all communities that began with 32-species will continuously contain 32-species, because of seed inflow. We thus report only the across-richness treatment models, omitting those within-richness treatments because they are invalid models.
Mirroring Shannon diversity in the manuscript, our models for the across-treatment effect were encoded as: Biomass ~ -1 + Stage + Stage:Richness. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.
The relationship between richness and total biomass was qualitatively similar to that of Shannon diversity. The direction and magnitude of the relationship between richness a total biomass are consistent across all of the models. The only variation between results is that in Forest2, the significance of the seed inflow and no seed inflow estimates both change; the seed inflow slope becomes significant, though maintaining almost no slope, and further the no seed inflow phase becomes insignificant, though again maintaining its general slope.
Again, the general structure of the relationship between the communities’ underlying coexistence dynamics and their emergent BEF relationship are maintained. The only qualitative difference is that the slope of the interaction is significantly less steep during the seed inflow phase. This results stems from richness not fully capturing the changes in species composition within the communities. Because seed addition ensures most species are likely present within the plots, richness during the seed inflow phase is unlikely to change.
This section of the document describes the statistical models’ validation, using richness as the focal biodiversity metric and total biomass as the focal ecosystem function.
Important terms:
Stage: With seed inflow, without seed inflowNinitial: Planted species richnessClark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 23.05 1.34 20.44 25.58 1.00 2568 2823
## StageWithoutseedinflow 6.16 1.68 2.93 9.35 1.00 2698 2605
## StageWithseedinflow:Richness 1.92 0.09 1.74 2.10 1.00 2826 2959
## StageWithoutseedinflow:Richness 7.01 0.31 6.43 7.63 1.00 2480 2691
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 18.59 0.47 17.73 19.51 1.00 3371 2494
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5453755 0.01619418 0.5126256 0.5761042
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 60.22 1.00 58.26 62.20 1.00 2843 2488
## StageWithoutseedinflow 60.29 0.99 58.37 62.25 1.00 3016 3112
## StageWithseedinflow:Richness 0.66 0.06 0.53 0.78 1.00 2870 2590
## StageWithoutseedinflow:Richness 0.87 0.06 0.74 1.00 1.00 3126 2945
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 13.85 0.36 13.18 14.59 1.00 3602 2826
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2835676 0.02358352 0.2353998 0.328111
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 50.10 0.67 48.79 51.42 1.00 2696 2857
## StageWithoutseedinflow 49.34 0.75 47.80 50.77 1.00 2161 2163
## StageWithseedinflow:Richness 0.26 0.05 0.17 0.35 1.00 2746 2683
## StageWithoutseedinflow:Richness 0.49 0.08 0.33 0.65 1.00 2244 2593
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.86 0.25 9.39 10.37 1.00 4096 2770
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.08429 0.01786053 0.05167383 0.1215699
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 42.31 1.55 39.23 45.34 1.00 2560 2202
## StageWithoutseedinflow 38.64 1.77 35.02 42.12 1.00 2516 2060
## StageWithseedinflow:Richness 1.21 0.14 0.94 1.49 1.00 2539 2337
## StageWithoutseedinflow:Richness 3.71 0.53 2.68 4.77 1.00 2349 2343
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 21.32 0.55 20.28 22.42 1.00 3509 2257
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1491274 0.02248032 0.1059508 0.194563
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 30.87 0.74 29.41 32.32 1.00 2504 2566
## StageWithoutseedinflow 23.48 0.94 21.59 25.32 1.00 2505 2478
## StageWithseedinflow:Richness -0.00 0.05 -0.10 0.09 1.00 2420 2441
## StageWithoutseedinflow:Richness -1.22 0.18 -1.56 -0.87 1.00 2533 2506
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 10.60 0.27 10.08 11.16 1.00 3811 2947
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2988578 0.02382471 0.2512006 0.344522
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.
Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.
A summary table of the BRMS model results:
## Family: gaussian
## Links: mu = identity; sigma = identity
## Formula: biomass ~ -1 + Stage + Stage:Richness
## Data: d_ (Number of observations: 770)
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup samples = 4000
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedinflow 76.05 0.65 74.74 77.29 1.00 2263 2480
## StageWithoutseedinflow 80.33 0.91 78.50 82.09 1.00 2316 2370
## StageWithseedinflow:Richness 0.04 0.04 -0.04 0.13 1.00 2353 2451
## StageWithoutseedinflow:Richness -1.80 0.30 -2.35 -1.21 1.00 2261 2434
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 9.19 0.24 8.74 9.65 1.00 3622 2474
##
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
Note the
Rhatsummary column: variation from 1.0 indicates the the model did not converge.
The Bayesian R-squared:
## Estimate Est.Error Q2.5 Q97.5
## R2 0.05147276 0.01445972 0.02542738 0.08153645
We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.
The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).
Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.
Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.